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Theorem llynlly 22851
Description: A locally 𝐴 space is n-locally 𝐴: the "n-locally" predicate is the weaker notion. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
llynlly (𝐽 ∈ Locally 𝐴 β†’ 𝐽 ∈ 𝑛-Locally 𝐴)

Proof of Theorem llynlly
Dummy variables 𝑒 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 llytop 22846 . 2 (𝐽 ∈ Locally 𝐴 β†’ 𝐽 ∈ Top)
2 llyi 22848 . . . . 5 ((𝐽 ∈ Locally 𝐴 ∧ π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) β†’ βˆƒπ‘’ ∈ 𝐽 (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝐽 β†Ύt 𝑒) ∈ 𝐴))
3 simpl1 1192 . . . . . . . 8 (((𝐽 ∈ Locally 𝐴 ∧ π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝐽 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝐽 β†Ύt 𝑒) ∈ 𝐴))) β†’ 𝐽 ∈ Locally 𝐴)
43, 1syl 17 . . . . . . 7 (((𝐽 ∈ Locally 𝐴 ∧ π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝐽 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝐽 β†Ύt 𝑒) ∈ 𝐴))) β†’ 𝐽 ∈ Top)
5 simprl 770 . . . . . . 7 (((𝐽 ∈ Locally 𝐴 ∧ π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝐽 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝐽 β†Ύt 𝑒) ∈ 𝐴))) β†’ 𝑒 ∈ 𝐽)
6 simprr2 1223 . . . . . . 7 (((𝐽 ∈ Locally 𝐴 ∧ π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝐽 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝐽 β†Ύt 𝑒) ∈ 𝐴))) β†’ 𝑦 ∈ 𝑒)
7 opnneip 22493 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑒 ∈ 𝐽 ∧ 𝑦 ∈ 𝑒) β†’ 𝑒 ∈ ((neiβ€˜π½)β€˜{𝑦}))
84, 5, 6, 7syl3anc 1372 . . . . . 6 (((𝐽 ∈ Locally 𝐴 ∧ π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝐽 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝐽 β†Ύt 𝑒) ∈ 𝐴))) β†’ 𝑒 ∈ ((neiβ€˜π½)β€˜{𝑦}))
9 simprr1 1222 . . . . . . 7 (((𝐽 ∈ Locally 𝐴 ∧ π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝐽 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝐽 β†Ύt 𝑒) ∈ 𝐴))) β†’ 𝑒 βŠ† π‘₯)
10 velpw 4569 . . . . . . 7 (𝑒 ∈ 𝒫 π‘₯ ↔ 𝑒 βŠ† π‘₯)
119, 10sylibr 233 . . . . . 6 (((𝐽 ∈ Locally 𝐴 ∧ π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝐽 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝐽 β†Ύt 𝑒) ∈ 𝐴))) β†’ 𝑒 ∈ 𝒫 π‘₯)
128, 11elind 4158 . . . . 5 (((𝐽 ∈ Locally 𝐴 ∧ π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝐽 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝐽 β†Ύt 𝑒) ∈ 𝐴))) β†’ 𝑒 ∈ (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘₯))
13 simprr3 1224 . . . . 5 (((𝐽 ∈ Locally 𝐴 ∧ π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (𝑒 ∈ 𝐽 ∧ (𝑒 βŠ† π‘₯ ∧ 𝑦 ∈ 𝑒 ∧ (𝐽 β†Ύt 𝑒) ∈ 𝐴))) β†’ (𝐽 β†Ύt 𝑒) ∈ 𝐴)
142, 12, 13reximssdv 3166 . . . 4 ((𝐽 ∈ Locally 𝐴 ∧ π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) β†’ βˆƒπ‘’ ∈ (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝐽 β†Ύt 𝑒) ∈ 𝐴)
15143expb 1121 . . 3 ((𝐽 ∈ Locally 𝐴 ∧ (π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯)) β†’ βˆƒπ‘’ ∈ (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝐽 β†Ύt 𝑒) ∈ 𝐴)
1615ralrimivva 3194 . 2 (𝐽 ∈ Locally 𝐴 β†’ βˆ€π‘₯ ∈ 𝐽 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝐽 β†Ύt 𝑒) ∈ 𝐴)
17 isnlly 22843 . 2 (𝐽 ∈ 𝑛-Locally 𝐴 ↔ (𝐽 ∈ Top ∧ βˆ€π‘₯ ∈ 𝐽 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝐽 β†Ύt 𝑒) ∈ 𝐴))
181, 16, 17sylanbrc 584 1 (𝐽 ∈ Locally 𝐴 β†’ 𝐽 ∈ 𝑛-Locally 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   ∈ wcel 2107  βˆ€wral 3061  βˆƒwrex 3070   ∩ cin 3913   βŠ† wss 3914  π’« cpw 4564  {csn 4590  β€˜cfv 6500  (class class class)co 7361   β†Ύt crest 17310  Topctop 22265  neicnei 22471  Locally clly 22838  π‘›-Locally cnlly 22839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-top 22266  df-nei 22472  df-lly 22840  df-nlly 22841
This theorem is referenced by:  llyssnlly  22852  efmndtmd  23475
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